Problem: An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6?
Answer: For any prime number, the sum of its proper divisors is equal to $1$, so a prime number cannot be an abundant number. Therefore, it suffices to check the smallest composite numbers that are not divisible by $6$. We find that:

$\bullet$ for $4$, $1 + 2 < 4$,
$\bullet$ for $8$, $1 + 2 + 4 < 8$,
$\bullet$ for $9$, $1 + 3 < 9$,
$\bullet$ for $10$, $1 + 2 + 5 < 10$,
$\bullet$ for $14$, $1 + 2 + 7< 14$,
$\bullet$ for $15$, $1 + 3 + 5< 15$,
$\bullet$ for $16$, $1 + 2 + 4 + 8 < 16$,
$\bullet$ for $20$, $1 + 2 + 4 + 5 + 10 = 22 > 20$.

Thus, the answer is $\boxed{20}$.